Wali's Number
Find the sum of all the digits of the number above when it is stated in decimal representation.
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2 solutions
For the sake of users who weren't familiar with that formula, they can even work out the problem as follows.
Observe that 9 9 9 . . . n 9 9 … 9 = = = 1 0 − 1 1 0 2 − 1 1 0 n − 1
Thus, we can write the given expression as 1 0 0 1 7 7 − 2 = 1 0 3 5 4 − 2 = ( 1 0 3 5 4 − 1 ) − 1 = 3 5 4 9 9 … 9 − 1 = 3 5 3 9 9 … 9 8
Thus sum of digits of above number would be 3 5 3 × 9 + 8 = 3 1 8 5
EDIT : Thus, you can even work out for the general formula of sum of digits of : 1 0 n − m where n ∈ N and 0 ≤ m ≤ 9 as follows 1 0 n − m = n 9 9 … 9 − m + 1 Thus, sum of digits will be 9 × ( n − 1 ) + ( 9 − ( m − 1 ) ) = 9 n − m + 1 S ( 1 0 n − m ) = 9 n − m + 1
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in response to @Kishlaya Jaiswal actually you did the same What I did but yes your answer is a detailed answer for those who want to know the concept of how I got the formula.Even I thought to upload the complete answer but I thought it would have been too long but it's good that u uploaded it. :)
See, 100^177 has (177 X 2) 0's = 354 0's and a 1. If you subtract 2, you get 353 9's and a 8. 353 X 9 = 3177 and added with 8 , it becomes 3185.
while practicing such type of question i found out the formula for solving the questions of the form 10^n-2 the formula is [(x-1)n-1] ;where x denotes 10 n denotes the power of 10 implies = 100^177-2={(10)^2}^177-2=10^354-2 now using my formula, (x-1)n-1=(10-1)354-1=3186-1=3185 if you want to understand how I deduced the formula then u can reply and I will be sending u the complete way of solving these type of questions. Hope it helped u to some extent. :)